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The coordinatewise uniformly Kadec–Klee property in some Banach spaces

Tao Zhang

Tongji University

Abstract: We introduce a new property $UKK_c$ for a Banach space and show for an Orlicz sequence space the following: $UKK_c\Leftrightarrow H_c \Leftrightarrow\Phi\in\delta_2$. Moreover, we show that the Orlicz-direct-sum spaces $\Bigl(\sum\limits_{n=1}^\infty\oplus X_n\Bigr)_{l_{\Phi}}$ and $\Bigr(\sum\limits_{n=1}^\infty\oplus X_n\Bigr)_{l_{(\Phi)}}$ have the property $H_c$ if every $X_n$ $(n\in\mathbb{N})$ has the property $H_c$ and $\Phi\in\delta_2$.

Keywords: Orlicz sequence space, Orlicz-direct-sum space, property $UKK_c$, property $H_c$.

UDC: 517.5

Received: 04.04.2002
Revised: 27.08.2002

 English version: Siberian Mathematical Journal, 2003, 44:2, 363–365

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